The limit of Dirichlet's Kernel

Question

Let's consider Dirichlet's Kernel for $\delta \le |t| \le \frac{1}{2}$. I am to prove that: $$|D_N(t)| \le \frac{1}{\sin (\pi \delta)}.$$ That is my attempt:
let's consider Dirichlet's Kernel written in the following form: $$D_N(t) = \frac{\sin\big(\pi(2N + 1)t \big)}{\sin(\pi t)}.$$ It's obvious that: $$|D_N(t)| \le \frac{1}{\sin(\pi t)} \tag{1}$$ because $|\sin x| \le 1, \forall_{x \in \mathbb{R}}$.
Now $|t| \le \frac{1}{2}$ and on the interval $[0, \frac{1}{2}]$ the function $\sin (\pi x)$ is increasing thus $(1)$ will be less than $\frac{1}{\sin (\pi \delta)}$.
Is my reasoning correct?